Question 124
Let's say that an ordered triple of positive integers $(a, b, c)$ is $n$-powerful if $a \leq b \leq c$, $\operatorname{gcd}(a, b, c)=1$, and $a^{n}+b^{n}+c^{n}$ is divisible by $a+b+c$. For example, $(1,2,2)$ is 5-powerful.
a) Determine all ordered triples (if any) which are $n$-powerful for all $n \geq 1$.
b) Determine all ordered triples (if any) which are 2004-powerful and 2005-powerful, but not 2007-powerful.
[Note that $\operatorname{gcd}(a, b, c)$ is the greatest common divisor of $a, b$ and $c$.]
a) Determine all ordered triples (if any) which are $n$-powerful for all $n \geq 1$.
b) Determine all ordered triples (if any) which are 2004-powerful and 2005-powerful, but not 2007-powerful.
[Note that $\operatorname{gcd}(a, b, c)$ is the greatest common divisor of $a, b$ and $c$.]